Probability in Catching Fish

In this discussion, you will use an Excel data set containing the "width percentage" of a group of perch, a type of fish.  The width percentage is found by multiplying the width by one hundred, then dividing the result by the length.  The mean of the data set is 16.18 and the standard deviation is 1.13. Using a java applet provided by Duxberry Press (©1999), we analyzed the probabilities for a normal distribution of this data set.

 The following two steps describe the steps taken using this applet. Many software packages and applets are available to do these computations. The discussion after these two steps illustrates the role that probability plays in making all statistical inference decisions. . 1. Fill in the mean, standard deviation, start and end points shown in figure 1.     *Note that the probability (Prob) = 0.95 for this data range. The probability is shown by the shaded area under the curve. This probability means that if you were catching fish from this population, 95 out of 100 fish would have width percentages within the range of 13.965 - 18.395. . 2. Enter a Start value of 0 and an End value of 13.965.  Now the area shaded under the curve has a probability of .025.  Now enter a Start value of 18.395 and an End value of 32.36 (any large number will work). The area under the curve again has a probability of .025. So, combining these two cases, if you were catching fish from this population of perch, only 5 out of 100 fish would have "width percentages" below 13.965 (very thin fish) or above 18.395 (very fat fish). In other words, a perch caught at random has only a 5% probability of an unusual width percentage (either very fat or very thin). .
 Probability is a key concept in statistical hypothesis testing. Let's say you are fishing and catch a fish with a width percentage of 13.3. Since it has an unusual width percentage for a perch, it seems unlikely this fish is a perch. It is too thin! Pretend this width percentage is the only information we have about the fish and we want to decide as best we can whether or not you caught a perch. The logic we use in hypothesis testing is as follows. This fish is either a perch or it is not a perch. Our null hypothesis is that the fish caught is a perch. Since the probability that a randomly caught perch would have an unusual width percentage is only 5%, we take this as strong evidence against the null being true. So, we reject the null and conclude our fish is not a perch (but we realize we might be mistaken). Note that we would also reject the null if the width percentage had been unusually large, for example 18.5. Finally, if the width percentage was in the usual range for a perch, say 15.3, then we would be unable to conclude anything about the fish. We simply have no evidence that the fish is not a perch. In hypothesis testing we say that we retain the null hypothesis. This doesn't show that the fish is a perch though.

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